Invariant metrics on current Lie algebras

Abstract

In this work we state conditions for a current Lie algebra $\mathfrak{g}\otimes \mathcal{S}$ to admit an invariant metric, where $\mathfrak{g}$ is a quadratic Lie algebra and $\mathcal{S}$ is an associative and commutative algebra with unit. We show that if $\mathfrak{g}$ is an indecomposable quadratic Lie algebra, then $\mathfrak{g}\otimes \mathcal{S}$ admits an invariant metric if and only if $\mathcal{S}$ also admits an invariant metric. In addition, we prove a theorem similar to the double extension for $\mathfrak{g}\otimes \mathcal{S}$, where $\mathfrak{g}$ is an indecomposable, nilpotent and quadratic Lie algebra.

KEYWORDS:

• Centroid
• commutative and associative algebra
• current Lie algebras
• invariant metric

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17A45
• 17B05
• Secondary: 15A63
• 17B60

Acknowledgments

The author also thanks to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.

Additional information

Funding

The author thanks the support provided by post-doctoral fellowship CONACYT 769309.